Estimating the sizes of hard-to-reach populations is important for many problems in public health and public policy. Population size estimation is particularly pressing in HIV/AIDS research because reliable estimates of the sizes of the most at-risk populations---drug injectors, female sex workers, and men who have sex with men---are critical for understanding and controlling the spread of the epidemic. Unfortunately, current statistical methods are not up to this challenge. The lack of timely and accurate information about the sizes of these most at-risk groups is a critical barrier to the design and evaluation of HIV prevention programs. The goal of this research is to improve the network scale-up method, a promising statistical approach for estimating the sizes of hard-to-reach groups. Network scale-up estimates come from survey data collected about the personal networks of a random sample of the general population, and offers important advantages over other approaches for estimating the sizes of hard-to-reach groups: 1) it can easily be standardized across countries and time because it requires a random sample of the general population, perhaps the most widely used sampling design in the world; 2) it can produce estimates of the sizes of many target populations in the same data collection, whereas many alternative methods require distinct data collections for each population of interest; and 3) it can be partially self-validating because it an easily be applied to populations of known size. However, despite these appeal characteristics and growing use by researchers and governments around the world, the statistical foundations of the scale-up method are poorly understood and key implementation questions remain unanswered. This research, which will be achieved through a combination of mathematical modeling, computer simulation, and the analysis of existing scale-up data sets, will enable researchers to collect more accurate and more useful information about hard-to-reach groups. Further, the statistical developments needed to achieve these aims will enrich our general ability to learn about complete networks from sampled data. Thus, this project combines foundational research about sampling in networks with important contributions to the global effort to contain the HIV/AIDS epidemic. PUBLIC HEALTH RELEVANCE: Estimating the sizes of hard-to-reach populations is important for many problems in public health and public policy. Population size estimation is particularly pressing in HIV/AIDS research because reliable estimates of the sizes of the most at-risk populations---drug injectors, female sex workers, and men who have sex with men---are critical for understanding and controlling the spread of the epidemic. The goal of this research is to improve the network scale-up method, a promising statistical approach for estimating the sizes of hard-to-reach groups.